Problems & Puzzles: Puzzles

Puzzle 3.- Magic Squares with consecutive primes

Harry Nelson was the first in produce (and he won a \$100.00 prize offered by Martin Gardner) a 3x3 matrix containing only consecutive primes.

His solution has in the central cell the prime 1480028171. The other cells has the following primes +/-12 ; +/-18, +/-30, +/-42.

(Ref. 2, p. 18)

While it was not easy produce the solutions, for sure you can assign the primes obtained by Nelson. Do you want to try ?

We know that Nelson got more than other 20 such magical squares.

1.- Is there any method for doing that ?

2.- Is there any 4x4 matrix with consecutive primes ?

Aale de Winkel sent (30/06/99) the following comment: :

"...given a regular order n magic square (numbers 1..n^2 : magic number t=n(n^2+1)/2) substitute each number (i) with a number (x + i * d) a magic square is obtained     with magic number nx+td = n(x+n^2+1)/2. hence any n^2 primes in arithmetic progression thus gives magic squares of order n."

According to this method you can get two 3x3 magic squares using the 10 consecutive primes in A.P. found by Dubner, Tony Forbes, Manfred Toplic et al.

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Several 4x4 prime magic squares can be found at: http://mathforum.org/te/exchange/hosted/suzuki/MagicSquare.4x4.prime.html

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Luke Pebody sent the following solutions to Q2 (April 2005):

31-101
67 89 43 59
47 79 101 31
83 37 41 97
61 53 73 71

37-103
79 59 71 67
53 73 61 89
97 101 41 37
47 43 103 83

79 67 83 47
41 101 61 73
59 71 43 103
97 37 89 53

1229-1321
1289 1279 1291 1259
1231 1283 1301 1303
1277 1237 1297 1307
1321 1319 1229 1249

1279 1289 1259 1291
1283 1303 1231 1301
1237 1277 1307 1297
1319 1249 1321 1229

1301 1231 1303 1283
1259 1291 1279 1289
1321 1319 1229 1249
1237 1277 1307 1297

4931-5021
4951 4931 5021 4993
4999 4967 4973 4957
5003 4987 4969 4937
4943 5011 4933 5009

12553-12689
12653 12577 12589 12659
12583 12611 12647 12637
12689 12619 12601 12569
12553 12671 12641 12613

3259909-3260063
3260051 3260017 3259933 3259979
3259999 3259931 3260063 3259987
3260021 3260029 3259957 3259973
3259909 3260003 3260027 3260041

3324329-3324521
3324389 3324521 3324407 3324361
3324353 3324457 3324509 3324359
3324499 3324371 3324341 3324467
3324437 3324329 3324421 3324491

26025107-26025281
26025211 26025239 26025127 26025257
26025203 26025179 26025199 26025253
26025271 26025229 26025227 26025107
26025149 26025187 26025281 26025217

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On October, 2005, Luke Pebody wrote:

Here is a 5x5 magic square

067 047 103 017 079
101 023 053 029 107
059 113 043 061 037
013 089 083 109 019
073 041 031 097 071

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Anurag Sahay wrote (July 07):

Some more 5x5 examples:

magic sum=313

19 29 113 43 109
103 83 13 67 47
53 71 59 107 23
101 41 31 79 61
37 89 97 17 73

103 59 67 13 71
47 17 89 53 107
31 79 73 101 29
113 97 43 37 23
19 61 41 109 83

I found an example with a different magic sum which is 703.
It contains consecutive primes starting from 79.

79 167 193 107 157
181 151 191 83 97
131 109 127 137 199
139 163 103 197 101
173 113 89 179 149

I also found another 5x5 example with a different magic sum: 785. It has consecutive primes starting with 97.

173 127 149 223 113
103 157 227 101 197
193 139 181 163 109
137 211 131 107 199
179 151 097 191 167

I found a 6x6 example . Magic sum=2316. Least prime=277

463 347 293 277 487 449
461 337 349 479 317 373
311 331 389 397 421 467
431 457 433 401 313 281
283 353 443 379 419 439
367 491 409 383 359 307

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On November 2012 Natalia Makarova wrote:

I looked puzzle number 3 on your site.
I have participated in this project.
In OEIS has an article:
https://oeis.org/A073520

I made a magic square of order 14 consecutive prime numbers (Magic Constant = 9660):

 89 97 193 227 241 521 563 683 911 1063 1231 1237 1283 1321 277 1091 1181 431 1153 757 739 937 941 433 1019 113 139 449 701 283 197 137 827 503 1069 1187 1087 233 499 877 1051 1009 1223 461 541 1259 149 1163 743 719 359 479 647 1291 163 463 311 883 601 239 1289 269 307 967 491 829 211 857 1109 1297 1097 809 1061 1129 823 487 523 587 443 733 199 853 317 599 617 977 167 643 127 709 773 509 577 811 997 971 1361 421 1301 439 1249 419 109 1279 1123 569 383 907 653 251 337 641 659 727 557 691 1327 401 983 179 1151 331 761 1117 157 619 229 991 607 389 257 859 1171 1213 547 1033 1319 631 263 151 1229 191 181 571 863 1307 281 293 1277 379 821 367 887 1013 673 919 929 1303 947 347 397 1039 173 1031 467 313 769 353 593 953 1093 1021 797 787 881 101 103 1049 223 409 457 1193 661 839 1103 1201 751 271 107 677 1217 349 613 373 1367 131

I also was a magic square of order 15, it is in this article OEIS:
https://oeis.org/A164843 ?

And there are a few articles in the drafting of the OEIS magic squares of prime numbers: A173981, A176571, A177434,  A188536,  A188537, A191533, A189188, A191679.

Here is the next larger one (15x15) gotten by Natalia Makarova, published in https://oeis.org/A073520 composed by 225 consecutive primes from 131 to 1619, with a magical sum constant of of 12669.

 131 167 229 461 541 617 733 911 967 1091 1259 1279 1319 1471 1493 547 907 1583 1613 149 1423 193 1601 941 137 233 389 1039 1283 631 1019 181 751 163 1453 1301 1297 1277 271 1619 1327 691 277 281 761 1307 719 359 919 1063 653 1237 269 1433 863 1439 313 191 1021 883 503 1367 433 1013 829 1153 317 347 1109 491 1249 677 1451 1489 241 421 311 1487 439 1049 1409 1123 463 409 983 449 1031 1163 373 1559 1399 1193 419 1531 971 647 977 1051 709 479 1229 379 353 1093 239 599 953 1213 587 499 727 1321 787 307 1151 157 1571 1033 773 991 211 1291 1499 577 1087 349 947 467 739 613 1171 1609 173 839 1097 563 139 1373 1459 1289 443 619 1201 1427 809 881 1303 331 263 569 607 1607 1511 673 1181 1481 1217 523 661 857 223 743 197 431 757 853 643 701 179 1483 571 769 859 1447 659 929 997 1223 1129 227 1549 887 257 557 367 1061 601 337 1361 937 1231 811 1543 293 877 1579 1187 397 1069 509 683 797 1567 401 383 641 283 823 827 1523 1381 1117 457 1429 199 151 521 1009 487 1597 251 593 1553 1103 821

One more received on December 2012: Stefano Tognon and Natalia Makarova, 35x35 magic square of consecutive primes for 3 to 9941, magic constant 163043, produced on year 2009. Bigger solutions here:http://digilander.libero.it/ice00/magic/prime/orderConstant.html

 1901 2003 647 2729 8627 9631 5407 7759 9929 2801 2999 499 5309 673 4651 5227 941 8951 1613 9791 7853 3617 9241 8269 1801 1723 3877 3677 1759 7109 2939 3413 2383 4547 8747 5081 787 4363 2767 5147 2029 619 6793 7573 1153 8087 2011 271 3593 3511 853 9473 9787 2281 7559 7151 157 7949 3217 7577 6779 4231 4987 7411 9413 2221 1889 9887 5927 509 757 2393 3 1181 3067 6353 3109 163 8009 2137 3041 5483 2777 1667 4451 9463 1619 1493 7523 2113 8699 8933 4051 9539 6733 2069 6883 7537 7541 2753 8527 1429 5623 8233 7691 8387 8419 433 3691 6827 3329 5507 4591 7793 1459 9619 7283 691 6961 5849 3853 571 2647 7127 6521 4019 7699 257 313 2027 2503 7753 2953 2833 2659 4271 3793 3389 7159 9857 7823 7057 239 1747 2243 2663 2381 6569 8147 3121 5281 1013 4583 557 1451 8123 7649 2239 6661 9679 569 4049 8839 7187 6427 5657 907 7477 2741 9283 1229 9127 5669 937 3719 5323 193 59 6899 1021 3709 8243 3833 8609 4421 827 8731 4289 233 5521 7927 8329 3251 8117 3607 37 5651 6469 2687 4409 5189 7687 2731 8111 1187 9371 4513 2837 4091 4931 661 7177 197 9461 2053 8689 2423 8647 631 4691 2711 367 8803 7951 6263 6947 6803 4639 5101 487 9323 8999 5303 719 8863 887 2161 2851 5393 2617 2857 23 8783 5351 4211 2591 9173 9349 1709 3779 7621 8209 3637 1471 6047 503 6737 9281 3457 607 6689 4003 3181 877 1499 5153 331 7757 4129 1063 2081 4153 5059 8221 9587 5237 419 7489 9467 2477 5009 7253 6323 2417 7669 3407 9181 9341 3079 4057 3313 8719 5279 1367 139 8807 9137 9811 3361 9391 3089 109 4549 709 3929 173 3889 2671 6871 4597 547 5171 1249 8167 263 9649 8849 593 7907 9497 2789 8693 9883 7211 3209 947 1993 6359 8059 1783 1033 8263 149 8539 31 6131 5023 179 11 3463 4673 107 8431 863 181 1811 7393 9151 577 9613 797 4241 7993 743 4993 4073 3541 6551 6299 9239 6977 6211 9293 6451 1609 2129 9521 2131 4079 7561 1481 9551 701 5903 8537 5693 79 4517 3461 1663 9419 2269 587 2927 1823 1031 5851 6421 2309 8543 8623 6337 137 4493 991 4933 9623 3371 3467 8707 6397 7247 3347 9013 1307 6073 53 1597 2971 2213 7309 5 5861 4663 6701 5791 1543 467 9067 6581 1319 1129 839 1201 5743 7297 4253 1093 8089 6761 1213 6203 421 8573 5051 8753 5233 6247 3643 5107 3221 3769 4621 9629 5231 2341 1097 7517 7207 4861 3229 7529 4201 4903 5519 3547 2963 9817 8971 5197 1151 3253 7883 379 6917 349 6089 3631 9157 2087 1193 3191 2549 2083 9479 4643 6113 3923 4243 281 7591 8017 1039 1549 131 4483 9743 6197 7717 7487 7817 2579 3911 3449 6763 3083 9319 1069 761 6959 8287 3931 167 9431 3623 2719 8501 7607 2677 5857 8563 4889 4723 967 5683 739 1091 4463 5653 3797 4391 9767 2543 5387 1877 9257 5477 683 821 4793 4133 7229 6563 3499 401 6833 6997 8821 4217 5639 9007 3917 9851 3559 9719 7039 2351 8923 1733 3169 3257 1399 1637 2819 211 5659 61 6121 1049 6199 1453 6101 8597 2089 7433 8039 2693 7901 6301 7013 2099 9091 1627 5179 4721 4283 2039 3767 7643 89 5471 953 3821 337 1009 8081 4507 4603 8377 5939 4397 4999 1531 7321 3539 229 6007 4919 8363 6163 2347 29 4817 3491 2521 8101 409 19 8941 8191 5879 4799 6133 643 2621 5099 5573 277 4327 8837 353 4679 6277 8677 4157 5953 5039 8093 4481 6599 3023 7121 3581 2957 7933 373 389 6653 3137 9161 4111 6547 4973 3533 983 5783 9739 8317 3203 971 6967 5479 1571 6043 8291 1741 1567 9547 7841 4259 4519 2593 2207 3701 4229 9697 5087 3391 7129 613 3469 7 9721 1559 2357 2251 1061 5821 9439 881 1087 5647 8713 9437 5779 9043 7681 7723 4177 9433 1861 1931 6317 3881 457 6691 3163 5897 359 1283 6991 2267 8963 3659 1301 1553 4021 4787 3259 8069 6949 6659 8273 2683 7069 2699 977 521 7349 6229 199 9907 7019 4139 1949 9311 1433 9277 1601 9833 4457 191 809 7417 1223 1669 9221 8629 7451 2437 4561 5827 811 4649 8179 5119 4027 2143 1237 6673 659 3823 2273 8761 6911 103 3061 1217 8219 7459 3583 7879 8741 2551 5717 9049 1933 3671 463 4337 6857 5441 4733 17 479 5527 7103 1621 4373 6091 6473 727 3803 7369 8443 4093 7583 9733 7877 3037 227 1231 3727 9001 1409 9901 5807 6173 4969 3343 653 5381 3613 3319 2909 6079 7603 8293 8423 2411 1327 443 7079 251 9941 2237 307 8861 4273 2879 4007 4261 2111 2713 5641 3967 7213 829 9769 8297 9109 9397 641 3863 4957 1787 1951 7829 2297 1997 8599 1321 6791 1303 2803 4357 6781 1297 8887 7333 8429 4909 5437 3119 5689 6011 1447 1279 1259 1171 4567 6491 6719 3271 1427 8581 8053 2153 6143 7547 3739 3761 3529 7789 1051 6449 1973 127 5399 6823 4099 6367 4759 4831 293 6379 6637 6373 563 3557 5443 1523 3187 5419 3527 6311 6067 9103 8641 1579 9677 997 439 1699 6271 9643 8681 97 911 9199 6679 6907 617 5113 4871 9931 4219 2377 1117 8969 7963 5843 8369 9059 7219 3019 8867 2389 7481 5569 1019 2609 2969 1697 919 1289 3947 67 2339 5557 2791 4339 8461 2539 5923 2063 43 8737 5563 3943 2467 5011 3373 2657 2203 4441 6529 7507 6257 6869 2311 4783 6607 5347 2399 8819 4127 1423 9871 8513 5813 9689 1583 241 5749 3919 7457 1979 2903 1847 5531 5413 6037 8831 3517 4967 4159 5333 3299 3571 9749 3011 8311 283 5801 859 523 7351 4877 7867 4297 1439 1163 151 7727 9859 7237 1693 8467 599 5869 6389 2689 3733 2441 73 1483 5503 7027 8389 5209 1657 1721 4751 8669 3851 3049 449 3307 4447 6829 5021 7193 2797 3359 113 9187 7703 1879 1489 311 6343 7589 4423 7549 773 8447 4013 9511 8929 3331 9803 9491 2557 2861 491 1291 2179 5591 1871 9203 9661 5741 9403 4937 5273 5987 9343 1831 6269 431 317 6703 5737 823 5261 7673 83 461 3907 6983 4729 6571 269 5981 5417 4523 8231 3697 1753 7499 7937 6863 6481 101 3167 41 2843 1777 769 7873 347 1511 1487 4703 6151 6287 8161 6577 6217 2333 857 6361 4813 8893 9133 8779 5867 2293 3301 9029 6029 4657 8011 5297 1867 2459 71 1913 1907 1123 2017 1987 751 6221 929 13 2141 5431 4349 6841 1103 2917 4951 9377 8171 9209 9923 1999 2473 7639 8521 9227 5449 9041 3847 1109 6553 7919 2887 9337 6709 2531 4943 9829 677 5839 6619 7307 9421 3989 883 2371 1873 397 2633 6053 3673 6329 1381 3433 8663 5881 4001 8353 1277 1607 9011 3001 2707 5077 47 3323 9533 5167 7043 8237 4637 5581 601 4801 4789 541 2749 7331 2897 383 223 1361 2447 7741 6971 7243 5701 5501 5711 9839 2287 9601 733 1789 7001 1373 9781 5003

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On July 01, 2016, Arkadiusz Wesolowski wrote:

Below are links to sequences giving magic sums of such magic squares of order 3, 4, 5, and 6.
http://oeis.org/A270305
http://oeis.org/A173981
http://oeis.org/A176571
http://oeis.org/A177434

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